Sketching Graphs using limit information

For each of the four cases below, sketch a graph of a function that satisfies the stated conditions. In each case, the domain of the function should be all real numbers. (professor also mentioned he wants us to write it out in piecemeal function format)

I think it is simpler than that though - awaiting confirmation from OP but:
- the function does not have to be named - no equation is asked for - just a sketch.
A random squiggle (provided each x only has one y) that fits the description will do.
The function does not even have to be continuous.

So, if a condition is that |f(x)|<a, then draw a dotted line at y=a and y=-a.
Another condition may be that f(p)=q ... then one places a dot at (x,y)=(p,q).
A line through the dot that does not touch the dotted lines will satisfy the conditions - leaves the limits.
I'd draw a vertical dotted line through the limit to help decide what happens there.

So.
a. what does it mean when ##\lim_{x\rightarrow a}f(x)\neq f(a)## ?
b. what does it mean that the limit does not exist?

c & d are cunning variations on a & b.

Mind you - being able to produce an equation would slam-dunk the question.

Anything where the left hand limit disagrees with the right-hand limit.
A step function will do. You can google "the limit does not exist" and look at pictures for examples.
There's also more on limits in the website I linked - please read the links people give you, we do that to save typing.

For the other thing:

|f(x)| < a just means that -a < f(x) < a.

eg. |sin(x)| < a is true as long as a > 1 :)
trivially, if f(x) is the horizontal line through the origin, then we can confidently use it as an example of |f(x)| < 2 .

In your sketch you draw a dotted, horizontal, line at y=a and y=-a and make sure that your wiggly line representing the sketched function stays between them.